Bayesian curve-fitting with radial basis functions under functional measurement error model

Abstract

Abstract This article presents Bayesian approach to regression splines with knots on a grid ofequally spaced sample quantiles of the independent variables under functional measure-ment error model. We consider small area model by using penalized splines of non-linearpattern. Speci cally, in a basis functions of the regression spline, we use radial basisfunctions. To t the model and estimate parameters we suggest a hierarchical Bayesianframework using Markov Chain Monte Carlo methodology. Furthermore, we illustratethe method in an application data. We check the convergence by a potential scale re-duction factor and we use the posterior predictive p value and the mean logarithmicconditional predictive ordinate to compar models.Keywords: Functional, hierarchical Bayes, measurement error, radial basis, semipara-metric. 1. Introduction We developed the semiparametric small area models with measurement errors models inour previous paper (Hwang and Kim, 2010). Speci cally, we considered small area model byusing penalized splines of non-linear pattern based on truncated polynomial basis functionsand knots on a grid of equally spaced sample quantiles under functional measurement errormodel. Measurement error modeling is also related to Goo and Kim (2013).The truncated polynomial basis functions (TPBF) is simple, but not always numericallystable when the number of knots is large and the smoothing parameter close to zero. In thiscase the computation has to be organized carefully and numerically superior alternativesare available, like B-splines and radial basis functions (Ruppert et al., 2003).The objective of this article is to develop alternative estimators of small area means byusing radial basis functions (RBF) with functional measurement error model. RBF is de nedwith degree pfrom Ruppert et al. (2003) as follows.1;x; ;x